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A163404 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%I A163404 #18 Jun 10 2025 20:52:03
%S A163404 1,11,110,1100,11000,109945,1098900,10983555,109781100,1097266500,
%T A163404 10967222970,109617836625,1095634704780,10950913128375,
%U A163404 109454819042250,1094005337374620,10934627535602100,109292043884611005
%N A163404 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
%C A163404 The initial terms coincide with those of A003953, although the two sequences are eventually different.
%C A163404 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163404 G. C. Greubel, <a href="/A163404/b163404.txt">Table of n, a(n) for n = 0..995</a>
%H A163404 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,9,9,9,-45).
%F A163404 G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
%F A163404 a(n) = 9*a(n-1)+9*a(n-2)+9*a(n-3)+9*a(n-4)-45*a(n-5). - _Wesley Ivan Hurt_, May 10 2021
%t A163404 CoefficientList[Series[(1 + x)*(1-x^5)/(1-10*x+54*x^5-45*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{9, 9, 9, 9, -45}, {1, 11, 110, 1100, 11000, 109945}, 30] (* _G. C. Greubel_, Dec 21 2016 *)
%t A163404 coxG[{5, 45, -9}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, May 12 2019 *)
%o A163404 (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6)) \\ _G. C. Greubel_, Dec 21 2016
%o A163404 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6) )); // _G. C. Greubel_, May 12 2019
%o A163404 (Sage) ((1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019
%K A163404 nonn,easy
%O A163404 0,2
%A A163404 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009